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point of rz'exe behind this ingenious recasting of the algebra of
complex numbers which are of importance for the sequel.
If as usual i denotes V  1, a 'complex number' is a number
of the type a -f bi, where a,b are 'real numbers' or, if preferred,
and more generally, elements of the field F defined by the above
postulates. Instead of regarding a + & as one ^tunber5,
Hamilton conceived it as an ordered couple of 'numbers', and he
designated this couple by writing it (a,b). He then proceeded to
impose definitions of sum and product on these couples, as
suggested by the formal rules of combination sublimated from
the experience of algebraists in manipulating complex numbers
as if the laws of common algebra did in fact hold for them.
One advantage of this new way of approaching complex
numbers was this: the definitions for sum and product of
couples were seen to be instances of the general, abstract
definitions of sum and product as in a field. Hence, if the con-
sistency of the system defined by the postulates for a field is
'proved, the like follows, without further proof, for complex
numbers and the usual rules by which they are combined. It
will be sufficient to state the definitions of sum and product in
Hamilton's theory of complex numbers considered as couples
(,&} (c4), etc.
The sum of (aj>) and (c,rf) is (a -f b, c + d); their product is
(ac  bd, ad ~- be). In the last, the minus sign is a's in a field;
namely, the element on postulated in IV is denoted by  a. To
the 0,1 of a field correspond here the couples (0,0), (1,0). With
these definitions it is easily verified that Hamilton's couples
satisfy all the stated postulates for a field. But they also accord
with the formal rules for manipulating complex numbers. Thus,
to (a.b). ic.d) correspond respectively a -f bi, c -j- di, and the
formal "sum" of these two is (a. -f c) -f i(b + <9 to ^bieh corre-
sponds the couple (a -f c, & -f d). Again, formal multiplication
of a -f bi, c -r- id gives (ac - bd) -f i(ad -f- be), to which cor-
responds the couple (ac - bd, ad + be). If this sort of thing is
new to any reader, it will repay a second inspection, as it is an
example of the way in which modem mathematics eliminates
mystery. So long as there is a shred of mystery attached to any
concept that concept is not mathematical.